diff --git a/executor/src/witgen/bus_accumulator/mod.rs b/executor/src/witgen/bus_accumulator/mod.rs
index c04359a826..db3edbc3ee 100644
--- a/executor/src/witgen/bus_accumulator/mod.rs
+++ b/executor/src/witgen/bus_accumulator/mod.rs
@@ -1,5 +1,5 @@
 use std::{
-    collections::{BTreeMap, BTreeSet},
+    collections::{BTreeMap, BTreeSet, HashSet},
     iter::once,
 };
 
@@ -15,6 +15,8 @@ use powdr_executor_utils::{
 use powdr_number::{DegreeType, FieldElement, KnownField};
 use rayon::iter::{IntoParallelRefIterator, ParallelIterator};
 
+use super::polynomial_references;
+
 mod extension_field;
 mod fp2;
 mod fp4;
@@ -120,20 +122,71 @@ impl<'a, T: FieldElement, Ext: ExtensionField<T> + Sync> BusAccumulatorGenerator
     }
 
     pub fn generate(&self) -> Vec<(String, Vec<T>)> {
+        let second_stage_cols = self
+            .pil
+            .committed_polys_in_source_order()
+            .filter(|(symbol, _)| symbol.stage == Some(1))
+            .flat_map(|(symbol, _)| symbol.array_elements())
+            .map(|(name, poly_id)| (poly_id, name))
+            .collect::<BTreeMap<_, _>>();
+        let intermediate_definitions = self.pil.intermediate_definitions();
+
         let accumulators = self
             .bus_interactions
             .par_iter()
             .flat_map(|bus_interaction| {
+                // Collect the PolyIDs of the accumulator columns.
+                let acc_cols = bus_interaction
+                    .accumulator_columns
+                    .iter()
+                    .map(|reference| {
+                        assert!(!reference.next);
+                        reference.poly_id
+                    })
+                    .collect::<Vec<_>>();
+                // Find the folded columns if they exist, by finding all second-stage columns that
+                // are referenced together with the accumulator columns.
+                let folded_cols = self
+                    .pil
+                    .identities
+                    .iter()
+                    .flat_map(|identity| match identity {
+                        Identity::Polynomial(_) => {
+                            let references =
+                                polynomial_references(identity, &intermediate_definitions)
+                                    .into_iter()
+                                    .filter(|col| second_stage_cols.contains_key(col))
+                                    .collect::<HashSet<_>>();
+                            if acc_cols.iter().any(|col| references.contains(col)) {
+                                references
+                                    .into_iter()
+                                    .filter(|col| !acc_cols.contains(col))
+                                    .collect()
+                            } else {
+                                vec![]
+                            }
+                        }
+                        _ => vec![],
+                    })
+                    .collect::<BTreeSet<_>>();
+                assert!(folded_cols.is_empty() || folded_cols.len() == acc_cols.len());
+
                 let (folded, acc) = self.interaction_columns(bus_interaction);
-                folded.into_iter().chain(acc).collect::<Vec<_>>()
+
+                assert!(folded_cols.is_empty() || folded_cols.len() == folded.len());
+                let acc = acc_cols.into_iter().zip_eq(acc);
+                let folded = folded_cols.into_iter().zip(folded);
+
+                acc.chain(folded).collect::<Vec<_>>()
             })
             .collect::<Vec<_>>();
 
-        self.pil
-            .committed_polys_in_source_order()
-            .filter(|(symbol, _)| symbol.stage == Some(1))
-            .flat_map(|(symbol, _)| symbol.array_elements().map(|(name, _)| name))
-            .zip_eq(accumulators)
+        accumulators
+            .into_iter()
+            .map(|(poly_id, acc)| {
+                let name = second_stage_cols.get(&poly_id).unwrap().clone();
+                (name, acc)
+            })
             .collect()
     }